Best Known (41, 53, s)-Nets in Base 8
(41, 53, 5463)-Net over F8 — Constructive and digital
Digital (41, 53, 5463)-net over F8, using
- 81 times duplication [i] based on digital (40, 52, 5463)-net over F8, using
- net defined by OOA [i] based on linear OOA(852, 5463, F8, 12, 12) (dual of [(5463, 12), 65504, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(852, 32778, F8, 12) (dual of [32778, 32726, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(852, 32779, F8, 12) (dual of [32779, 32727, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(841, 32768, F8, 10) (dual of [32768, 32727, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(852, 32779, F8, 12) (dual of [32779, 32727, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(852, 32778, F8, 12) (dual of [32778, 32726, 13]-code), using
- net defined by OOA [i] based on linear OOA(852, 5463, F8, 12, 12) (dual of [(5463, 12), 65504, 13]-NRT-code), using
(41, 53, 32127)-Net over F8 — Digital
Digital (41, 53, 32127)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(853, 32127, F8, 12) (dual of [32127, 32074, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(853, 32781, F8, 12) (dual of [32781, 32728, 13]-code), using
- construction XX applied to Ce(11) ⊂ Ce(9) ⊂ Ce(8) [i] based on
- linear OA(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(841, 32768, F8, 10) (dual of [32768, 32727, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(81, 12, F8, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(11) ⊂ Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(853, 32781, F8, 12) (dual of [32781, 32728, 13]-code), using
(41, 53, large)-Net in Base 8 — Upper bound on s
There is no (41, 53, large)-net in base 8, because
- 10 times m-reduction [i] would yield (41, 43, large)-net in base 8, but